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Properties

Label 435.2.q.d
Level $435$
Weight $2$
Character orbit 435.q
Analytic conductor $3.473$
Analytic rank $0$
Dimension $36$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [435,2,Mod(41,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("435.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 435.q (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.47349248793\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{2} + 2 q^{3} + 36 q^{5} - 8 q^{6} + 8 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{2} + 2 q^{3} + 36 q^{5} - 8 q^{6} + 8 q^{7} - 4 q^{8} + 4 q^{10} + 12 q^{11} + 10 q^{12} - 28 q^{14} + 2 q^{15} - 60 q^{16} + 20 q^{17} + 32 q^{18} + 16 q^{19} - 12 q^{21} + 24 q^{24} + 36 q^{25} - 4 q^{26} - 22 q^{27} + 28 q^{29} - 8 q^{30} - 8 q^{31} + 16 q^{32} + 8 q^{33} + 8 q^{35} - 28 q^{36} - 4 q^{37} - 24 q^{38} - 24 q^{39} - 4 q^{40} - 48 q^{41} + 8 q^{42} + 4 q^{43} - 16 q^{44} + 20 q^{46} + 20 q^{47} - 66 q^{48} + 28 q^{49} + 4 q^{50} - 44 q^{52} - 24 q^{54} + 12 q^{55} + 84 q^{56} - 28 q^{57} - 64 q^{58} + 10 q^{60} + 20 q^{61} - 8 q^{62} - 32 q^{63} - 8 q^{66} - 60 q^{68} - 36 q^{69} - 28 q^{70} + 16 q^{71} + 64 q^{72} + 8 q^{73} + 2 q^{75} + 16 q^{76} - 32 q^{77} + 48 q^{78} + 12 q^{79} - 60 q^{80} - 60 q^{81} + 56 q^{82} + 100 q^{84} + 20 q^{85} - 8 q^{86} - 10 q^{87} - 24 q^{88} - 20 q^{89} + 32 q^{90} + 16 q^{92} - 24 q^{93} + 52 q^{94} + 16 q^{95} + 8 q^{96} + 4 q^{97} + 8 q^{98} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −1.93705 + 1.93705i 1.50032 − 0.865476i − 5.50431i 1.00000 −1.22972 + 4.58265i 1.59227 6.78801 + 6.78801i 1.50190 − 2.59698i −1.93705 + 1.93705i
41.2 −1.82725 + 1.82725i −0.389527 + 1.68768i − 4.67770i 1.00000 −2.37206 − 3.79558i 1.47752 4.89284 + 4.89284i −2.69654 − 1.31480i −1.82725 + 1.82725i
41.3 −1.34488 + 1.34488i 1.17986 − 1.26804i − 1.61741i 1.00000 0.118585 + 3.29214i −4.79407 −0.514537 − 0.514537i −0.215844 − 2.99223i −1.34488 + 1.34488i
41.4 −1.05880 + 1.05880i 1.69213 + 0.369716i − 0.242102i 1.00000 −2.18308 + 1.40017i 1.34177 −1.86126 − 1.86126i 2.72662 + 1.25122i −1.05880 + 1.05880i
41.5 −1.00351 + 1.00351i −1.62379 + 0.602757i − 0.0140454i 1.00000 1.02461 − 2.23435i 4.09600 −1.99292 − 1.99292i 2.27337 − 1.95750i −1.00351 + 1.00351i
41.6 −0.780332 + 0.780332i −0.476047 + 1.66535i 0.782165i 1.00000 −0.928048 − 1.67100i 0.394466 −2.17101 − 2.17101i −2.54676 − 1.58557i −0.780332 + 0.780332i
41.7 −0.602690 + 0.602690i −1.20419 − 1.24496i 1.27353i 1.00000 1.47608 + 0.0245761i 0.173231 −1.97292 − 1.97292i −0.0998697 + 2.99834i −0.602690 + 0.602690i
41.8 −0.0236294 + 0.0236294i 1.03028 + 1.39231i 1.99888i 1.00000 −0.0572444 − 0.00855463i 3.42666 −0.0944913 − 0.0944913i −0.877056 + 2.86893i −0.0236294 + 0.0236294i
41.9 0.203088 − 0.203088i 0.407389 − 1.68346i 1.91751i 1.00000 −0.259155 − 0.424627i 4.71926 0.795601 + 0.795601i −2.66807 − 1.37164i 0.203088 − 0.203088i
41.10 0.264256 − 0.264256i 1.60964 − 0.639569i 1.86034i 1.00000 0.256348 − 0.594368i −1.21376 1.02012 + 1.02012i 2.18190 − 2.05896i 0.264256 − 0.264256i
41.11 0.398500 − 0.398500i 0.654884 + 1.60347i 1.68239i 1.00000 0.899957 + 0.378013i −4.31678 1.46744 + 1.46744i −2.14225 + 2.10018i 0.398500 − 0.398500i
41.12 0.602660 − 0.602660i −1.60971 + 0.639395i 1.27360i 1.00000 −0.584771 + 1.35545i −0.244261 1.97287 + 1.97287i 2.18235 − 2.05849i 0.602660 − 0.602660i
41.13 0.998333 − 0.998333i −1.31362 − 1.12890i 0.00666265i 1.00000 −2.43844 + 0.184414i 2.30961 2.00332 + 2.00332i 0.451185 + 2.96588i 0.998333 − 0.998333i
41.14 1.39967 − 1.39967i −1.39384 + 1.02821i − 1.91814i 1.00000 −0.511758 + 3.39006i −0.338767 0.114580 + 0.114580i 0.885569 − 2.86632i 1.39967 − 1.39967i
41.15 1.48187 − 1.48187i 0.154682 + 1.72513i − 2.39186i 1.00000 2.78563 + 2.32719i 2.93783 −0.580678 − 0.580678i −2.95215 + 0.533693i 1.48187 − 1.48187i
41.16 1.48848 − 1.48848i 1.06719 − 1.36422i − 2.43117i 1.00000 −0.442117 − 3.61912i −1.31013 −0.641785 − 0.641785i −0.722191 − 2.91178i 1.48848 − 1.48848i
41.17 1.85345 − 1.85345i −1.61221 − 0.633081i − 4.87058i 1.00000 −4.16154 + 1.81476i −4.54343 −5.32050 − 5.32050i 2.19842 + 2.04131i 1.85345 − 1.85345i
41.18 1.88783 − 1.88783i 1.32654 + 1.11368i − 5.12777i 1.00000 4.60672 − 0.401834i −1.70742 −5.90468 − 5.90468i 0.519415 + 2.95469i 1.88783 − 1.88783i
191.1 −1.93705 − 1.93705i 1.50032 + 0.865476i 5.50431i 1.00000 −1.22972 − 4.58265i 1.59227 6.78801 − 6.78801i 1.50190 + 2.59698i −1.93705 − 1.93705i
191.2 −1.82725 − 1.82725i −0.389527 − 1.68768i 4.67770i 1.00000 −2.37206 + 3.79558i 1.47752 4.89284 − 4.89284i −2.69654 + 1.31480i −1.82725 − 1.82725i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
87.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 435.2.q.d yes 36
3.b odd 2 1 435.2.q.c ✓ 36
29.c odd 4 1 435.2.q.c ✓ 36
87.f even 4 1 inner 435.2.q.d yes 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.q.c ✓ 36 3.b odd 2 1
435.2.q.c ✓ 36 29.c odd 4 1
435.2.q.d yes 36 1.a even 1 1 trivial
435.2.q.d yes 36 87.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 4 T_{2}^{35} + 8 T_{2}^{34} - 4 T_{2}^{33} + 125 T_{2}^{32} - 500 T_{2}^{31} + 1008 T_{2}^{30} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(435, [\chi])\). Copy content Toggle raw display